Optimal control of a reaction-diffusion epidemic model with non-compliance

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

Turnpike results hold a central position in the area of applied mathematics, offering profound insights into the long-time behavior of optimal control systems. They play an important role in guiding decision-making processes in dynamic systems. Often dynamic systems depend on uncertain data and/or parameters. In this paper, optimal boundary control problems for the transport equation with random initial data and random source term are considered. The paper begins with the numerical analysis of the turnpike phenomenon for an optimal boundary control problem with nonlinear time-varying feedback control, where the initial data are perturbed by randomized Fourier series and the source term is perturbed by a random variable. Further, an integral turnpike result is presented for the optimal controls and the corresponding optimal states of an optimal boundary control problem with random initial data. A turnpike result is also presented for an optimal boundary control problem governed by a transport equation with random source term, where optimal control, optimal expected state, and variance satisfy an integral turnpike inequality. The turnpike constant, the expected value, and the variance are further specified for Gaussian and uniformly distributed random variables.

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

In this paper, an iterative adaptive dynamic programming (ADP) algorithm is developed to solve the optimal cooperative control problems for residential multi-battery systems. To avoid solving high-dimensional optimal control problems, we first constrain all the batteries at their worst performance, which transforms the multi-input optimal control problem into a single-input one. Based on the worst-performance optimal control law, the optimal cooperative control law for the residential multi-battery systems is obtained, where in each iteration, only a single-input optimization problem is implemented. Finally, numerical results are given to illustrate the performance of the developed algorithm.

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

An optimal birth control problem for a dynamical population model with size-structure

We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a nonlinear model derived from a semi-linear approximation of the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. This involves a p-Laplace-type problem on the graph with $$p=\frac {3}{2}$$ . We prove well-posedness, existence of optimal controls and derive a first order optimality condition.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

In this paper, we consider a class of optimal switching control problems with multiple time-delays and a cost on changing control and subject to terminal state constraints. A computational method involving three stages is developed to solve this class of optimal control problems. First, by parameterizing the control function with piecewise-constant functions, the optimal switching control problem is approximated by a sequence of finite-dimensional optimization problems, where the original switching times, the control heights and the control switching times are decision variables. Second, by introducing new variables, the total variation of the control variables is transformed into an equivalently smooth function. Third, we convert the constrained optimization problem into one only with box constraints by an exact penalty function method. The gradients of the cost functional are then derived, which can be combined with any gradient-based optimization method to determine the optimal solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

This article tackles the optimal control design problem for second order nonlinear affine systems using Krotov sufficient conditions. The computation of optimal control law(s) for nonlinear systems is usually done using the iterative methods based on the standard tools of optimal control theory which viz. Calculus of Variations (CoV), Hamilton-Jacobi-Bellman equation, Pontryagin's principle, etc. This work utilizes the Krotov sufficient conditions of global optimality to obtain non-iterative solutions. These conditions are derived by transforming the optimal control problem into another equivalent optimization problem. This translation is done via an ad-hoc selection of the so-called Krotov function. In this article, the Krotov function is chosen such that the equivalent optimization problem is solved non-iteratively to obtain optimal and sub-optimal control laws for the original optimal control problem. The proposed methodology is demonstrated by a numerical example.

A stochastic minimax optimal time-delay state feedback control strategy for uncertain quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal state feedback control problem of uncertain quasi-integrable Hamiltonian system with time delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed state feedback control forces are approximated by the control forces without time delay and the original problem is converted into a stochastic optimal state feedback control problem of an uncertain system without time delay. After that, by following a procedure based on the stochastic averaging method and stochastic differential game, the worst-case disturbances and the optimal controls are obtained from solving a Hamiltonian–Jacobi–Isaacs (HJI) equation. As an example, the stochastic minimax optimal state feedback control of a Duffing oscillator with parametric disturbances and time-delayed feedback control is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.

Optimal Design and Control of Cyclically Operated Biological Wastewater Treatment Plants

The optimal control problem is investigated for oscillation processes, described by integrodifferential equations with the Fredholm operator when functions of external and boundary sources nonlinearly depend on components of optimal vector controls. Optimality conditions having specific properties in the case of vector controls were found. A sufficient condition is established for unique solvability of the nonlinear optimization problem and its complete solution is constructed in the form of optimal control, an optimal process, and a minimum value of the functional.

Decomposition method for solving two optimal control problems and one optimization problem in batch fermentation is proposed. The problems are formulated based on a nonstructured mathematical model with slowly varying parameters and a finite cost criterion of maximum end production. Dependence of the model parameters on one physical or chemical parameter, which could easily be used as a control input is introduced analytically in the model equations and three model descriptions are obtained by nonlinear difference equations. Sensitivity functions of state trajectories towards slowly varying coefficients are introduced to account for model uncertainties. Based on them extended optimal control and optimization problems are formulated.